Integrand size = 22, antiderivative size = 49 \[ \int \frac {x}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \sqrt {b c-a d}} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {455, 65, 214} \[ \int \frac {x}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \sqrt {b c-a d}} \]
[In]
[Out]
Rule 65
Rule 214
Rule 455
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^2\right ) \\ & = \frac {\text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{d} \\ & = -\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \sqrt {b c-a d}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98 \[ \int \frac {x}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{\sqrt {b} \sqrt {-b c+a d}} \]
[In]
[Out]
Time = 2.92 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.78
method | result | size |
pseudoelliptic | \(\frac {\arctan \left (\frac {b \sqrt {d \,x^{2}+c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}\) | \(38\) |
default | \(-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x -\frac {\sqrt {-a b}}{b}\right )^{2}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {d \left (x +\frac {\sqrt {-a b}}{b}\right )^{2}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}\) | \(300\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (39) = 78\).
Time = 0.28 (sec) , antiderivative size = 231, normalized size of antiderivative = 4.71 \[ \int \frac {x}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\left [\frac {\log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right )}{4 \, \sqrt {b^{2} c - a b d}}, -\frac {\sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (b^{2} c - a b d\right )}}\right ] \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (42) = 84\).
Time = 3.11 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.73 \[ \int \frac {x}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\begin {cases} \frac {\operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b \sqrt {\frac {a d - b c}{b}}} & \text {for}\: d \neq 0 \\\begin {cases} \frac {x^{2}}{2 a \sqrt {c}} & \text {for}\: b = 0 \\\tilde {\infty } x^{2} & \text {for}\: \sqrt {c} = 0 \\\frac {\log {\left (2 a \sqrt {c} + 2 b \sqrt {c} x^{2} \right )}}{2 b \sqrt {c}} & \text {otherwise} \end {cases} & \text {otherwise} \end {cases} \]
[In]
[Out]
Exception generated. \[ \int \frac {x}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d}} \]
[In]
[Out]
Time = 5.30 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.80 \[ \int \frac {x}{\left (a+b x^2\right ) \sqrt {c+d x^2}} \, dx=\frac {\mathrm {atan}\left (\frac {b\,\sqrt {d\,x^2+c}}{\sqrt {a\,b\,d-b^2\,c}}\right )}{\sqrt {a\,b\,d-b^2\,c}} \]
[In]
[Out]